Friday, August 13, 2010

compositionality and the predicate abstraction rule

The semantic account given in Heim and Kratzer (1998) is not strictly compositional and the failure is very similar to the composition failure in Kaplan below. The failure in this case is due to their Predicate Abstraction Rule (seeHeim and Kratzer 1998, p. 186). According to their theory there are lambda terms in the object language syntax such as 'λx.Fx' but there is not a lexical entry for the lambda binder ‘λx’ itself. And the semantic value of 'λx.Fx' isn’t (and can't) be calculated by composing the semantic value of ‘λx’ with the semantic value of ‘Fx’ but instead such lambda terms must be handled by the following non-compositional rule.

Why not instead have the semantic value of `Fx' be a function from assignments to extensions -- and the semantic value of `λx' be a function from functions from assignments to extensions to functions from individuals to extensions? Then the semantic value of `λx.Fx' can be calculated from the semantic values of `λx' and `Fx' as follows.

[[λx.Fx]]^g = [[λx]]^g (λg.[[Fx]]^g) = λi. [[Fx]]^g[i/x]

To see the similarity to the discussion of Kaplan's LD below notice that we could add a non-compositional rule to patch up the semantics of Kaplan's LD.

This would allow semantic calculations of the relevant constructions but would be explicitly non-compositional.

Why would one opt for having these non-compositional rules when the system can easily be transformed into a compositional system with no such non-compositional rules? Is it that much of a cost to have a few non-compositional rules like the predicate abstraction rule?

In the discussions of temporalism/eternalism in relation to the operator argument why don't the eternalists just respond by claiming that temporal constructions need special non-compositional rules?

Past Tense Construction Rule: Let α be a branching node with daughters β and γ, where β dominates only the temporal operator PAST. Then, for any time t, [[α]]^c,t = ∀t' before t. [[γ]]^t'.

In this case it seems that the eternalist theory with a few non-compositional tense rules is a bad theory. But if we are happy with the Predicate Abstraction Rule why should a few more such rules really bother us?

I guess what I have ended up doing was pushing some kind of slippery slope worry -- predicate abstraction is a gateway rule that will lead to nasty ones like the past tense construction rule. That is probably not a very threatening argument. But I just don't understand why we would opt for predicate abstraction in the first place when a truly compositional semantics is close at hand.